arXiv subjects

Yuehui Zhang

Publications and source records attributed to Yuehui Zhang.

7 recordsLinked to original sources

arXiv2024

Quasi-Hereditary Orderings of Nakayama Algebras

Let $A$ be an algebra with iso-class of simple modules $\mathcal{S}$ of cardinality $n$. A total ordering on $\mathcal{S}$ making every Weyl module Schurian and every indecomposable projective module filtered by the Weyl modules is called to be a quasi-hereditary ordering or $q$-ordering on $A$ and $A$ is a quasi-hereditary algebra under this ordering. The number of $q$-orderings on $A$ is denoted by $q(A)$. To determine whether an ordering on $\mathcal{S}$ is a $q$-ordering is a hard problem. A famous result due to Dlab and Ringel is that $A$ is hereditary if and only if every ordering is a $q$-ordering, equivalently, $q(A)=n!$. The twenty-years old $q$-ordering conjecture claims that $q(A)\le\dfrac{2}{3}n!$. The present paper proves a very simple criterion for $q$-orderings when $A$ is a Nakayama algebra. This criterion is applied to getting a full classification of all $q$-orderings of $A$ and an explicit iteration formula for $q(A)$, and also a positive proof of the $q$-ordering conjecture for Nakayama algebras.

math.RT↗

arXiv2021

Nilpotent Category of Abelian Category and Self-Adjoint Functors

Let $\mathcal{C}$ be an additive category. The nilpotent category $\mathrm{Nil} (\mathcal{C})$ of $\mathcal{C}$, consists of objects pairs $(X, x)$ with $X\in\mathcal{C}, x\in\mathrm{End}_{\mathcal{C}}(X)$ such that $x^n=0$ for some positive integer $n$, and a morphism $f:(X, x)\rightarrow (Y,y)$ is $f\in \mathrm{Hom}_{\mathcal{C}}(X, Y)$ satisfying $fx=yf$. A general theory of $\mathrm{Nil}(\mathcal{C})$ is established and it is abelian in the case that $\mathcal{C}$ is abelian. Two abelian categories are equivalent if and only if their nilpotent categories are equivalent, which generalizes a Song, Wu, and Zhang's result. As an application, it is proved all self-adjoint functors are naturally isomorphic to $\mathrm{Hom}$ and $\mathrm{Tensor}$ functors over the category $\mathrm{Nil}$ of finite-dimensional vector spaces. Both $\mathrm{Hom}$ and $\mathrm{Tensor}$ can be naturally generalized to $\mathrm{HOM}$ and $\mathrm{Tensor}$ functor over $\mathrm{Nil}(\mathcal{V})$. They are still self-adjoint, but intrinsically different.

math.CT↗

arXiv2021

A Generalized Determinant of Matrices and Applications

A generalized definition of the determinant of matrices is given, which is compatible with the usual determinant for square matrices and keeps many important properties, such as being an alternating multilinear function, keeping multiplication formula and partly keeping the Cauchy-Binet's formula. As applications of the new theory, the generalized Cramer's rule and the generalized oriented volume are obtained.

math.CA↗

arXiv2016

Unbounded ladders induced by Gorenstein algebras

The derived category $D({\rm Mod}A)$ of a Gorenstein triangular matrix algebra $A$ admits an unbounded ladder; and this ladder restricts to $D^-({\rm Mod})$ {\rm(}resp. $D^b({\rm Mod})$, $D^b({\rm mod})$, $K^b({\rm proj})${\rm)}. A left recollement of triangulated categories with Serre functors sits in a ladder of period $1$; as an application, the singularity category of $A$ admits a ladder of period $1$.

math.RT↗

arXiv2016

Unbounded ladders induced by Gorenstein algebras

The derived category of a Gorenstein triangular matrix algebra $A$ admits an unbounded ladder, which is of period $3$ if $A = T_2(B)$. Also, a left recollement of triangulated categories with Serre functors sits in a ladder of period $1$; as an application, the singularity category of $A$ admits a ladder of period $1$.

math.RT↗

arXiv2013

Injective Objects of Monomorphism Categories

For an acyclic quiver $Q$ and a finite-dimensional algebra $A$, we give a unified form of the indecomposable injective objects in the monomorphism category ${\rm Mon}(Q,A)$ and prove that ${\rm Mon}(Q, A)$ has enough injective objects. As applications, we show that for a given self-injective algebra $A$, a tilting object in the stable category $\underline{A}$-mod induces a natural tilting object in the stable monomorphism category $\underline{\rm Mon}(Q,A)$. We also realize the singularity category of the algebra $kQ\otimes_k A$ as the stable monomorphism category of the module category of $A$.

math.RT↗

arXiv2013

Repetitive cluster-tilted algebras

Let $H$ be a finite dimensional hereditary algebra over an algebraically closed field $k$ and $\mathscr{C}_{F^m}$ be the repetitive cluster category of $H$ with $m\geq 1$. We investigate the properties of cluster tilting objects in $\mathscr{C}_{F^m}$ and the structure of repetitive cluster-tilted algebras. Moreover, we generalized Theorem 4.2 in \cite{bmrrt} (Buan A, Marsh R, Reiten I. Cluster-tilted algebra. Trans. Amer. Math. Soc., 359(1)(2007), 323-332.) to the situation of $\mathscr{C}_{F^m}$, and prove that the tilting graph $\mathscr{K}_{\mathscr{C}_{F^m}}$ of $\mathscr{C}_{F^m}$ is connected.

math.RT↗